### Graduate Student Seminar, 4:30 pm May 2, 2005; HH127

#### Speaker:

Jacob Heidenreich
#### Title:

The Great Change: Mathematics at the Turn of the (Last) Century
#### Abstract:

In the late 19th and early 20th centuries, a great shift occured in
mathematics. Mathematicians thought differently about what the goals
and techniques of mathematics should be. Before the shift, research
mathematics looked like a complex version of undergraduate Calculus
(what's the deriviative of x^2 - x at the point...). After the shift,
research mathematics looked like a complex version of graduate Real
Analysis (let X be a measure space and S be the sigma algebra of
measurable subsets...). Mathematics changed from being about something
to being about nothing. Specifically, it changed from being about
particular things (solve *this* equation) to being about nothing
*in particular* (anything that satisfies a certain set of axioms). My
talk will study this shift, the mathematical advances which motivated
it, and the many arguments and name-callings it gave rise to ("That's
not mathematics, that's theology!"-- Gordon). In particular, we'll
look at the Hilbert-Frege debate over what mathematical axioms and
definitions are. Frege argued that a statement is considered an axiom
if its truth is beyond doubt (think of the Euclidean Axioms) and we
should call a statement an axiom only if we're certain of it. Hilbert
said an axiom set was really a definition of a kind of mathematical
structures (think of the group theory axioms as defining what a group
is) and that this is how a mathematician should think of *any* axiom
set, including the Euclidean Axioms. Frege said Euclidean Geometry is
"about" the Euclidean plane, and Hilbert said Euclidean Geometry is
"about" any structure that satisfies the Euclidean axioms -- we must
not say what. As Russell put it, "Mathematics is the subject in which
we don't know what we are talking about, nor whether what we are saying
is true."

To volunteer to give a talk, or for any other questions regarding this schedule,
contact Sara Miller